45  Reproducing Kernel Hilbert Space

Author
Affiliation

Aditya Mahajan

McGill University

Updated

May 7, 2024

45.1 Review of Linear Operators

Linear Operator

Let \(\mathcal F\) and \(\mathcal G\) be normed vector spaces over \(\reals\). A function \(A \colon \mathcal F \to \mathcal G\) is called a linear operator if it satisfies the following properties:

  • Honogeneity: For any \(α \in \reals\) and \(f \in \mathcal F\), \(A(αf) = α (Af)\).
  • Additivity: For any \(f,g \in \mathcal F\), \(A(f + g) = Af + Ag\).

The operator norm of a linear operator is defined as \[ \NORM{A} = \sup_{f \in \mathcal F} \frac{ \NORM{A f}_{\mathcal G}} {\NORM{f}_{\mathcal F}}. \]

If \(\NORM{A} < ∞\), then the operator is said to be a bounded operator.

As an example, suppose \(\mathcal F\) is an inner product space. For a \(g \in \mathcal F\), the operator \(A_g \colon \mathcal F \to \reals\) defined by \(A_g(f) = \langle f, g \rangle\) is a linear operator. Such scalar valued operators are called functionals on \(\mathcal F\).

Linear operators satisfy the following property.

Theorem 45.1 If \(A \colon \mathcal F \to \mathcal G\) is a linear operator, then the following three conditions are equivalent:

  1. \(A\) is a bounded operator.
  2. \(A\) is continuous on \(\mathcal F\).
  3. \(A\) is continious at one point of \(\mathcal F\).

45.2 Dual of a linear operator

There are two notions of dual of a linear operator: algebraic dual and topological dual. If \(\mathcal F\) is a normed space, then the space of all linear functionals \(A \colon \mathcal F \to \reals\) is the algebraic dual space of \(\mathcal F\); the space of all continuous linear functions \(A \colon \mathcal F \to \reals\) is the topological dual space of \(\mathcal F\).

In finite-dimensional space, the two notions of dual spaces coincide (every linear operator on a normed, finite dimensional space is bounded). But this is not the case for infinite dimensional spaces.

Theorem 45.2 (Riesz representation) In a Hilbert space \(\mathcal F\), all continuous linear functionals are of the form \(\langle\cdot, g\rangle\), for some \(g \in \mathcal F\).

Two Hilbert spaces \(\mathcal F\) and \(\mathcal G\) are said to be isometrically isomorphic if there is a linear bijective map \(U \colon \mathcal F \to \mathcal G\) which preserves the inner product, i.e., \(\langle f_1, f_2 \rangle_{\mathcal F} = \langle U f_1, U f_2 \rangle_{\mathcal G}\).

Note that Riesz representation theorem gives a natural isometric isomorphism \(\psi \colon g \mapsto \langle \cdot, g \rangle_{\mathcal F}\) between \(\mathcal F\) and its topological dual \(\mathcal F'\), whereby \(\NORM{ψ(g)}_{\mathcal F'} = \NORM{g}_{\mathcal F}\).

45.3 Reproducing kernel Hilbert space

Let \(\mathcal H\) be a Hilbert space of functions mapping from some non-empty set \(\ALPHABET X\) to \(\reals\). Note that for every \(x \in \ALPHABET X\), there is a very special functional on \(\mathcal H\): the one that assigns to each \(f \in \mathcal H\), its value at \(x\). This is called the evaluation functional and denoted by \(δ_x\). In particular, \(δ_x \colon \mathcal H \to \reals\), where \(δ_x \colon f \mapsto f(x)\).

Reproducing kernel Hilbert space (RKHS)

A Hilbert space \(\mathcal H\) of functions \(f \colon \ALPHABET X \to \reals\) defined on a non-empty set \(\ALPHABET X\) is said to be a RKHS if \(δ_x\) is continuous for all \(x \in \ALPHABET X\).

In view of Theorem 45.1, an equivalent definition is that a Hilbert space \(\mathcal H\) is RKHS if the evaluation functionals \(δ_x\) are bounded, i.e., for every \(x \in \ALPHABET X\), there exists a \(M_x\) such that \[ | δ_x | = | f(x) | \le M_x \| f \|_{\mathcal H}, \quad \forall f \in \mathcal H\]

An immediate implication of the above property is that two functions which agree in RKHS norm agree at every point: \[ | f(x) - g(x) | = | δ_x(f - g) | \le M_x \| f - g \|_{\mathcal H}, \quad \forall f,g \in \mathcal H. \]

For example, the \(L_2\) space of square integrable functions i.e., \(\int_{\reals^n} f(x)^2 dx < ∞\) with inner product \(\int_{\reals^n} f(x) g(x)dx\) is a Hilbert space, but not an RKHS because the delta function, which has the reproducing property \[ f(x) = \int_{\reals^n} δ(x - y) f(y) dy \] is not bounded.

RKHS are particularly well behaved. In particular, if we have a sequence of functions \(\{f_n\}_{n \ge 1}\) which converges to a limit \(f\) in the Hilbert-space norm, i.e., \(\lim_{n \to ∞} \NORM{f_n - f}_{\mathcal H} = 0\), then they also converge pointwise, i.e., \(\lim_{n \to ∞} f_n(x) = f(x)\) for all \(x \in \ALPHABET X\).

45.4 Properties of RHKS

RKHS has many useful properties:

  1. For any RKHS, there exists a unique kernel \(k \colon \ALPHABET X × \ALPHABET X \to \reals\) such that

    • for any \(x \in \ALPHABET X\), \(k(\cdot, x) \in \mathcal H\),
    • for any \(x \in \ALPHABET X\) and \(f \in \mathcal H\), \(\langle f, k(\cdot, x) \rangle = f(x)\) (the reproducing property).

    In particular, for any \(x,y \in \ALPHABET X\), \[ k(x,y) = \langle k(\cdot, x), k(\cdot, y) \rangle. \] Thus, the kernel is a symmetric function.

  2. The kernel is positive definite, i.e., for any \(n \ge 1\), for all \((a_1, \dots, a_n) \in \reals^n\) and \((x_1, \dots, x_n) \in \ALPHABET X^n\), \[ \sum_{i=1}^n \sum_{j=1}^n a_i a_i h(x_i, x_j) \ge 0 \]

    A conseuqence of positive definiteness is that \[| k(x, y)|^2 \le k(x, x) k(y, y). \]

  3. (Moore-Aronszajn Theorem) For every positive definite kernel \(K\) on \(\ALPHABET X × \ALPHABET X\), there is a unique RKHS on \(\ALPHABET X\) with \(K\) as its reproducing kernel.

45.5 Examples of kernels

Some common examples of symmetric positive definite kernels for \(\ALPHABET X = \reals^n\) are as follows:

  1. Linear kernel \[ k(x,y) = \langle x, y \rangle\]

  2. Gaussian kernel \[ k(x,y) = \exp\biggl( - \frac{\| x - y \|^2}{σ^2} \biggr), \quad σ > 0. \]

  3. Polynomail kernel \[ k(x,y) = \bigl( 1 + \langle x, y \rangle \bigr)^d, \quad d \in \integers_{> 0}. \]

45.6 Properties of kernels

  1. Suppose \(φ \colon \ALPHABET X \to \reals^n\) is a feature map, then \[ k(x,y) := \langle φ(x), φ(y) \rangle \] is a kernel.

    Note that there are no conditions on \(\ALPHABET X\) (e.g., \(\ALPHABET X\) doesn’t need to be an inner product space).

  2. If \(k\) is a kernel on \(\ALPHABET X\), then for any \(α > 0\), \(αk\) is also a kernel.

  3. If \(k_1\) and \(k_2\) are kernels on \(\ALPHABET X\), then \(k_1 + k_2\) is also a kernel.

  4. If \(\ALPHABET X\) and \(\ALPHABET Y\) be arbitrary sets and \(A \colon \ALPHABET X \to \ALPHABET Y\) is a map. Let \(k\) be a kernel on \(\ALPHABET Y\). Then, \(k(A(x_1), A(x_2))\) is a kernel on \(\ALPHABET X\).

  5. If \(k_1 \colon \ALPHABET X_1 × \ALPHABET X_1 \to \reals\) is a kernel on \(\ALPHABET X_1\) and \(k_2 \colon \ALPHABET X_2 × \ALPHABET X_2 \to \reals\) is a kernel on \(\ALPHABET X_2\), then \[ k( (x_1, x_2), (y_1, y_2) ) = k_1(x_1, y_1) k_2(x_2, y_2) \] is a kernel on \(\ALPHABET X_1 × \ALPHABET X_2\).

  6. (Mercer-Hilber-Schmit theorems) If \(k\) is positive definite kernel (that is continous with finite trace), then there exists an infinite sequence of eiegenfunctions \(\{ φ_i \colon \ALPHABET X \to \reals \}_{i \ge 1}\) and real eigenvalues \(\{λ_i\}_{i \ge 1}\) such that we can write \(k\) as: \[ k(x,y) = \sum_{i=1}^∞ λ_i φ_i(x) φ_i(y). \] This is analogous to the expression of a matrix in terms of its eigenvector and eigenvalues, except in this case we have functions and an infinity of them.

    Using this property, we can define the inner product of RKHS in a simpler form. First, for any \(f \in \mathcal H\), define \[ f_i = \langle f, φ_i \rangle.\] Then, for any \(f, g \in \mathcal H\), \[ \langle f, g \rangle = \sum_{i=1}^∞ \frac{ f_i g_i } { λ_i }. \]

45.7 Kernel ridge regression

Given labelled data \(\{ (x_i, y_i) \}_{i=1}^n\), and a feature map \(φ \colon \ALPHABET X \to \ALPHABET Z\), define the RKHS \(\ALPHABET H\) of functions from \(\ALPHABET Z \to \reals\) with the kernel \(k(x,y) = \langle φ(x), φ(y) \rangle_{\mathcal H}\). Now, consider the problem of minimizing

\[f^* = \arg \min_{f \in \ALPHABET H} \biggl( \sum_{i=1}^n \bigl( y_i - \langle f, φ(x_i) \rangle_{\mathcal{H}} \bigr)^2 + λ \NORM{f}^2_{\mathcal H} \bigr).\]

Theorem 45.3 (The representer theoreom (simple version)) Given a loss function \(\ell \colon \ALPHABET Z^n \to \reals\) and a penalty function \(Ω \colon \reals \to \reals\), there is as a solution of \[ f^* = \arg \min_{f \in \mathcal H} \ell(f(x_1), \dots, f(x_n)) + Ω(\NORM{f}^2_{\mathcal H}). \] that takes the the form \[ f^* = \sum_{i=1}^n α_i k(\cdot, x_i).\]

If \(Ω\) is strictly increasing, all solutions have this form.

Using the representer theorem, we know that the solution is of the form \[ f = \sum_{i=1}^n α_i φ(x_i). \] Then, \[ \sum_{i=1}^n \bigl( y_i - \langle f, φ_i(x_i) \rangle_{\mathcal H} \bigr)^2 + λ \NORM{f}_{\mathcal H}^2 = \NORM{ y - K α}^2 + λ α^\TRANS K α. \]

Differentiating wrt \(α\) and setting this to zero, we get \[ α^* = (K + λI_n)^{-1} y. \]